Following solution will drop specific default constraint of a column from the table
Declare @Const NVARCHAR(256)
SET @Const = (
SELECT TOP 1 'ALTER TABLE' + YOUR TABLE NAME +' DROP CONSTRAINT '+name
FROM Sys.default_constraints A
JOIN sysconstraints B on A.parent_object_id = B.id
WHERE id = OBJECT_ID('YOUR TABLE NAME')
AND COL_NAME(id, colid)='COLUMN NAME'
AND OBJECTPROPERTY(constid,'IsDefaultCnst')=1
)
EXEC (@Const)